Basic question about discrete minimal surfaces

Question

Let $P$ be a convex polygon with $n > 3$ vertices $v_1, \ldots, v_n \in \mathbb{R}^2$, let $x$ be a point in the interior of $P$, and let $u$ be a function with prescribed values at the vertices of $P$, $u( v_{i} ) = u_{i} \in \mathbb{R}$ for $i = 1, \ldots, n$. Assume that the points $( v_{i}, u_{i} ) \in \mathbb{R}^{3}$ are not coplanar. For what value of $u(x)$ does the (piecewise linear) graph of $u$ over $P$ have minimal area? How does the solution relate to the geometric median of the points $( v_{i}, u_{i} )$?

Answer 1

For any interior edge of your area minimizing surface the sum of all 4 adjusted angles has to be at least $\pi$. Check my paper "Area minimizing polyhedral surfaces are saddle" (Sorry for self-advertizement.)

Answer 2

A few remarks:

(1) One interpretation of what is meant by "the (piecewise linear) graph of $u$ over $P$" is a triangulation of the 3D polygon determined by $(v_i,u_i)$.

(2) In general, determining whether or not a 3D polygon has a triangulation (not all 3D polygons are triangulable) is NP-hard. Your class is restricted in that its projection to 2D is a convex polygon. And all your polygons are triangulable, e.g., by triangulating in 2D and lifting to 3D.

(3) If self-intersections among the triangles are permitted, then there is an $O(n^4)$ algorithm to find a triangulation that minimizes the sum of triangle areas.

(4) There are practical algorithms implemented, e.g.,

Zou, Ming, Tao Ju, and Nathan Carr. "Delaunay-restricted Optimal Triangulation of 3D Polygons." (2012). (PDF download.)

This relatively recent paper summarizes much of the earlier literature.

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                  ^{(Fig.9 detail from Zou & Carr.)}

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